Properties

Label 8-1840e4-1.1-c1e4-0-5
Degree $8$
Conductor $1.146\times 10^{13}$
Sign $1$
Analytic cond. $46599.3$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·9-s + 18·11-s − 14·19-s + 10·25-s + 12·29-s + 22·31-s − 18·41-s + 49-s − 24·59-s − 2·61-s − 6·71-s − 4·79-s + 44·81-s + 12·89-s + 162·99-s − 24·101-s + 10·109-s + 161·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 3·9-s + 5.42·11-s − 3.21·19-s + 2·25-s + 2.22·29-s + 3.95·31-s − 2.81·41-s + 1/7·49-s − 3.12·59-s − 0.256·61-s − 0.712·71-s − 0.450·79-s + 44/9·81-s + 1.27·89-s + 16.2·99-s − 2.38·101-s + 0.957·109-s + 14.6·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(46599.3\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.20287718\)
\(L(\frac12)\) \(\approx\) \(14.20287718\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$D_4\times C_2$ \( 1 - p^{2} T^{2} + 37 T^{4} - p^{4} T^{6} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
11$C_4$ \( ( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 45 T^{2} + 833 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 33 T^{2} + 569 T^{4} - 33 p^{2} T^{6} + p^{4} T^{8} \)
19$C_4$ \( ( 1 + 7 T + 39 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_4$ \( ( 1 - 11 T + 81 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 40 T^{2} + 1518 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 9 T + 101 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$$\times$$C_2^2$ \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \)
47$D_4\times C_2$ \( 1 - 48 T^{2} + 494 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
61$D_{4}$ \( ( 1 + T + 111 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 240 T^{2} + 23198 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 3 T + 143 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 36 T^{2} - 538 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 240 T^{2} + 27998 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 39 T^{2} + 17297 T^{4} + 39 p^{2} T^{6} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.66581620143539781578983484820, −6.29720852306682266490098189385, −6.29179340046815438745998539320, −6.28413194147013180876171839497, −6.19887520711151453972069857347, −5.51036736273608631414599207964, −5.21023822604839441202975886528, −4.64626836929443240811067997655, −4.62806068679460302769009863625, −4.58689221081780799182260439810, −4.34425528069808022455527823395, −4.32487028003837155697461268750, −4.16650434119763488993331408300, −3.78083280748645323801804237318, −3.63151586006120526617734760802, −3.15955995242963240899943542471, −3.04674386223366010725449921372, −2.85545714268729118782634492363, −2.24311497210516514057745001151, −1.72797408067733927151121193433, −1.67960189489757914005721342674, −1.64866083417210551987370985058, −1.16496266223136356532337259221, −0.894142865497532192362774522542, −0.77518004565332744831613777496, 0.77518004565332744831613777496, 0.894142865497532192362774522542, 1.16496266223136356532337259221, 1.64866083417210551987370985058, 1.67960189489757914005721342674, 1.72797408067733927151121193433, 2.24311497210516514057745001151, 2.85545714268729118782634492363, 3.04674386223366010725449921372, 3.15955995242963240899943542471, 3.63151586006120526617734760802, 3.78083280748645323801804237318, 4.16650434119763488993331408300, 4.32487028003837155697461268750, 4.34425528069808022455527823395, 4.58689221081780799182260439810, 4.62806068679460302769009863625, 4.64626836929443240811067997655, 5.21023822604839441202975886528, 5.51036736273608631414599207964, 6.19887520711151453972069857347, 6.28413194147013180876171839497, 6.29179340046815438745998539320, 6.29720852306682266490098189385, 6.66581620143539781578983484820

Graph of the $Z$-function along the critical line